Are neural networks invertible?
The meaning of invertible here is the standard definition of invertibility for a mathematical function $f \colon X \to Y$. Invertible simply means &quot;the function has an inverse map $f^{-1} \colon Y \to X$&quot;. Equivalently the function $f$ is bijective, which means the following two conditions hold:

$f$ is injective: for any two distinct $x_1, x_2 \in X$, $f(x_1) \ne f(x_2)$.

$f$ is surjective: for any $y \in Y$, there exists an $x \in X$ such that $f(x) = y$.


If this is unfamiliar, you should be able to find some helpful references on Google using these terms.
Most obvious neural network architectures cannot possibly be invertible. Consider for example a classifier which takes an image or some other high-dimensional input, and outputs a classification label. This network could only be invertible if there was only one possible input1 which corresponds to each label, which is not the goal of the network.

1 I mean this rather literally: if one pixel is even slightly different, this would be a different input to the network and would have to have a different output.